Fermat's Last Theorem

It has all the makings of a great mystery - a 17th century genius, an ancient Greek text, and a 10 year old boy, who in the 1960s was determined to solve the mystery of a 350 year old maths problem - Fermat's last theorem.

TRANSCRIPT

Simon PampenaIn mathematics, greatness is measured by the problems you solve, and 20 years ago one person solved one problem that made them immortal. That person was Andrew Wiles. The problem was called Fermat's Last Theorem, and the story of how he solved it is inspirational. So, to celebrate the anniversary, I've invited a new generation of maths prodigies to hear and help tell the story.

Michelle Chen Wanting to do something like prove Fermat's Last Theorem, it's very ambitious.

Anand BharadwajFor several hundred years the great minds like Gauss, Euler, they tried to attack this question but they had no clue where to start.

Simon PampenaHands up if you've spent more than a day on a maths problem. A month. A year! Well, there's one guy that beats everyone. It's this guy here. Andrew Wiles.

NARRATIONWiles was born in the United Kingdom in 1953, but the path his life would take was paved much earlier.

Simon PampenaIt all started in the 17th century when amateur mathematician and genius Pierre de Fermat was thumbing through his copy of an ancient text, the Arithmetica. The book contained mathematics that was centuries old, but in the margins Fermat wrote equations that were new - astonishingly new. But it wasn't until his death that anyone actually discovered any of this great work, and he never left a proof for any of these awesome discoveries. That was left for the reader to work out for themselves. It was if he'd, like, published his own Lonely Planet guide to maths. And so what he was doing was saying 'Hey, guys, take this book and you can find out all these great places you can go and visit'. But there was no directions, there was no way to get there. So people worked out how to get there, they made up their own proofs. And all of the notes got proofs except one. It was the last one. It was Fermat's Last Theorem.

Matthew CheahPeople started to doubt whether it was actually true, and others thought that maybe Fermat made a mistake.

Alexander GunningThe best mathematicians tended to think that proof was not going to be accessible any time soon, because there was no obvious way of attacking the problem, no way of saying, 'OK, I'm going to apply specialised techniques 'X' to this problem.'

Anand BharadwajAnd after a few hundred years they just abandoned the question and went for something more mainstream.

NARRATIONAt least until a young Andrew Wiles discovered it in the 1960s.

Alan GuoAndrew Wiles was ten years old and he was in a local library, and he found a maths book with Fermat's Last Theorem inside it and he was just really intrigued by this question, because he could understand it, and from then on he made it his dream to attack this problem and eventually find a solution.

NARRATIONSo what was Fermat's Last Theorem? Well, it begins with something we're all taught at school, Pythagoras's Theorem. So, Pythagoras' Theorem state A squared plus B squared equals C squared.And Fermat's Last Theorem extends that, swapping the squared into any whole number bigger than two. It could be three, four, five, any number all the way to infinity. And there'll be no possible whole-number solutions to that equation.

Simon PampenaIt's odd. Why should that be the case? That's a pattern. There should be a reason for that. So, anyway, he says 'I'm gonna try and solve it', but the problem was the world that he was dealing with, the world that was described in Fermat's Lonely Planet, that had changed.

NARRATIONIn the 1950s mathematics had become very strange. Fermat's Lonely Planet guide was now just a chapter in a much larger book of discoveries, The Hitchhiker's Guide To Euclidian Space.

Simon PampenaAnd worse still, mathematicians had found other spaces that were so radically different from one another that they needed Hitchhiker's Guides of their own.

NARRATIONIn the pages of The Hitchhiker's Guide To Projective Space existed objects that were embedded on the surface of spheres, called elliptic curves, while The Hitchhiker's Guide To Hyperbolic Space detailed objects that walked away with negative curvature that you couldn't even visualise. These were called modular forms.

Simon PampenaAnd so modern mathematics didn't seem to have anything to do with Fermat's Last Theorem.

NARRATIONBut the eventual answer to Fermat's Last Theorem came from just such an unlikely connection between the two radically different spaces. Two Japanese mathematicians, Taniyama and Shimura, made a suggestion that no-one believed. They said that elliptic curve using whole numbers or fractions was actually a modular form in disguise. But what the heck did that mean?

Simon PampenaWell, elliptic curves are a lot like the seam on a tennis ball. They're created by projecting a curve from the plane onto a sphere so that the points at infinity meet up to form a closed loop. But a modular form is a completely different beast. Being modular means that there's a regular pattern that extends out in all directions of hyperbolic space. To get your head around it, in regular space, it looks like this. One bit that's actually repeated indefinitely. So what were Taniyama and Shimura actually saying?

NARRATIONThey were suggesting a connection between the two objects. They said, without bending or warping you could always slot the infinite pattern of a modular form through the closed loop of an elliptic curve - much like a coil can slot through a circle - but only if the elliptic curves were generated by whole numbers or fractions.

Simon PampenaSo, what did this have to do with Fermat's Last Theorem? Nothing. It wasn't until a long time later that this guy came along. Frey. I love this guy, Frey, because he came into Taniyama and Shimura in the '80s and he changed everything. Frey considered the unthinkable. What if Fermat was wrong? What if you could find whole numbers that fit the equation for powers higher than two? Well, if such a solution existed, Frey created an elliptic curve that turned out not to be modular. But Taniyama and Shimura said every elliptic curve generated by whole numbers had to be modular. So that meant there was a link between the two. If Fermat's Last Theorem was wrong, so was Taniyama and Shimura. But if Taniyama and Shimura was right then you'd finally prove Fermat's Last Theorem. And, of course, who was interested in that? This guy. Andrew Wiles was ecstatic. Finally he had a way to actually prove his childhood dream. He converted a 350-year-old problem to only a 30-year-old problem.

Michelle CheBasically, Wiles decided that he would start working on the Taniyama-Shimura conjecture in order to prove Fermat's Last Theorem.

Alan GuoHe worked in secret and without his peers knowing.

Michelle CheHe was working on the same problem for seven years, not knowing that he would even necessarily be able to solve it.

Yong See FooThat would be something just out of someone's mind to do.

Anand BharadwajIt's just an astonishing level of commitment.

Simon PampenaSo how did he do it? Well, if I could tell you in a few sentences, it wouldn't have taken him seven years. But basically it was a counting argument. He was able to work out how to convert elliptic curves into modular forms and vice versa, but he didn't know whether he had the same amount after doing that process, so he counted them. But that's hard to do 'cause there's infinitely many of them. And that's what he worked out. He worked out a way to actually count infinite sets and he got there. Wow.

Anand BharadwajIt's just amazing.

Matthew CheahTo have worked on it for so long, I really think that he deserved to archive his childhood dream.

Alexander GunningHe's one of the few mathematicians who's managed to solve something famously hard which is obviously what we'd like to do if we could manage to do it, which is probably not gonna happen but it's worth a shot.

Simon PampenaSo he solved his problem, Fermat's Last Theorem was finally solved. But don't worry, there's still unsolved problems that you could solve, so get cracking and solve some problems, because immortality awaits, people.